Theorem fmptapd 7170

Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)

Hypotheses

Ref	Expression
fmptapd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fmptapd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fmptapd.s	⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
fmptapd.c	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)

Assertion

Ref	Expression
fmptapd	⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥

Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem fmptapd

Step	Hyp	Ref	Expression
1		fmptapd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)
2		fmptapd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fmptapd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4	1, 2, 3	fmptsnd 7168	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
5	4	uneq2d 4115	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
6		mptun 6679	. . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
7	6	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
8		fmptapd.s	. . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
9	8	mpteq1d 5195	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶))
10	5, 7, 9	3eqtr2d 2801	1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∪ cun 3897  {csn 4584  ⟨cop 4590   ↦ cmpt 5186

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-mpt 5187

This theorem is used by:  fmptpr  7171  poimirlem3  38373  poimirlem4  38374  poimirlem16  38386  poimirlem17  38387  poimirlem19  38389  poimirlem20  38390